# Limit calculator

One of the basic concepts of mathematical analysis is the limit of a function - the value to which the value of the function (ƒ) tends as the argument tends to point x.

The limit of the function ƒ(x) can be expressed as the limit of a sequence of values: ƒ(x1), ƒ(x2), ƒ(x3), ..., corresponding to the sequence of elements of the domain of definition of the function x1, x2, x3, .. In the case where finding the limit is possible, the function converges, that is, it has a finite limit. If there is no limit, the function diverges.

A function is called continuous when at one of the points (from the range of values) the limit is equal to the value of the function. To indicate the method of convergence of a function, a special base of subsets is introduced, a special case of which is a system of punctured neighborhoods. The function is denoted by ƒ, the point by x, and the limit of the function by A or lim.

The theory of limits is the broadest category of mathematical analysis: there are dozens of different techniques and nuances for finding the desired limits. Moreover, they were all invented in a relatively short period of time, because the very concept of “function limit” appeared no more than 400 years ago.

## History of origin and development

The prerequisites for finding the limits of functions arose back in Antiquity. Thus, Archimedes, among other studies, used the passage to the limit (taking the limit) to calculate the volumes of geometric figures, which, in fact, can be attributed to finding the limits of functions.

But the very concept of “limit” arose much later - at the turn of the 17th-18th centuries - thanks to the outstanding English scientist Isaac Newton. In addition to him, the passage to the limit was also used by John Wallis, which was reflected in his scientific work “Arithmetic of Infinite Quantities.”

Although the concept of “function limit” was not the basis of integral and differential calculus, it was used for a long time as an applied (implicit) quantity that does not have a clear definition.

In one form or another, limits can be found in the works of Leonhard Euler and Joseph Louis Lagrange, dating back to the mid-17th century. But the final definition of the limit of a function was formulated only at the beginning of the 19th century - by the Czech scientist Bernard Bolzano and the French mathematician Augustin Louis Cauchy. The first introduced a strict definition of the sequence limit in 1816, and the second in 1821.

In the 19th century, the theory of limits began to be used in the analysis of infinite series of numbers, using it to justify the rules of mathematical analysis. According to the definition of Augustin Louis Cauchy: “If the values successively assigned to the same variable approach a fixed value indefinitely, so that in the end they differ from it as little as possible, then the latter is called the limit of all others.”

In 1861, the German mathematician Karl Weierstrass introduced the modern notation for limits: lim and lim(x→0), and introduced epsilon-delta as it is used in mathematics today. The main purpose of limits was and remains to carry out analyzes of infinite series, as well as to create new functions with their help. 400 years after the start of using limits, they have become in demand in many branches of science and natural history, including those used in modern high technologies.

Today, limits describe economic, chemical, biological and sociological processes, allow you to create accurate forecasts and study changes in interdependent quantities in dynamics. The limit of a function is compatible with simple functions as well as logarithmic, exponential, and trigonometric functions. It can also be used to describe and calculate quantities in the vicinity of infinity.

Using existing formulas for calculating limits, it is possible to carry out mathematical analysis with high accuracy and in an abbreviated form of notation. If without using the limit of a function, describing a particular process could take several pages of text, with the introduction of this concept, even the most complex process of integration/differentiation can be described with a single formula.